Learning, macroeconomic dynamics and the term structure: A structural econometric model

1
Learning, macroeconomic dynamics and the term
structure A structural econometric model

• Hans Dewachter and Marco Lyrio
• University of Leuven and Rotterdam School of
  Management
• Warwick Business School

2
Overview

• Motivation
• A macroeconomic model of the term structure
• Standard New-Keynesian model
• Learning
• Affine time-varying term-structure model
• Estimation
• Methodology
• Data
• Results
• Cross-validation based on survey expectations
• Conclusions and further steps

3
Motivation

• The main motivation for this research is the
  observation that standard rational expectations
  models cannot account for the term-structure
  dynamics (puzzle).
• Standard RE models should describe the
  term-structure accurately
• Given that standard RE models are found to model
  macroeconomic dynamics reasonably.
• Given that policy rates can be modeled in terms
  of Taylor rules, relating the policy rate to
  macroeconomic variables.
• Assuming no-arbitrage conditions hold in bond
  markets.
• Assuming consistent market prices of risk
• However, they do not. They fail significantly at
  the long end of the term structure.

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Motivation Term structure fit RE
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Motivation Term structure fit VAR
6
Motivation Fitting long-term yields. Literature
review

• VAR approach (Ang and Piazzesi, 2003) include
  additional latent factors without explicit
  macroeconomic interpretation.
• Stochastic endpoint approach (Kozicki and
  Tinsley, 2001, Dewachter and Lyrio, 2005)
  include latent factors with an interpretation of
  a stochastic endpoint (of inflation)
• However no explanation as to why long-run
  inflation expectations/ inflation targets move.
• Structural approach (Hördahl et al, 2005, Bekaert
  et al, 2005, Rudebusch and Wu, 2005) include
  additional latent factors with a macroeconomic
  interpretation based on a structural (RE) model
  (long-run inflation target).
• However no explanation as to why long-run
  inflation expectations/inflation targets move.

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Motivation Introducing learning dynamics

• The main aim of this paper is to extend the
  standard RE model in order to account for the
  yield curve dynamics without reference to latent
  factors.
• We use a standard New-Keynesian structural model
  as benchmark (as in Bekaert et al, 2005 or
  Hördahl et al, 2005).
• By introducing learning, we allow for additional
  factors, i.e. (subjective) market expectations.
• The model thus combines the actual observed
  macroeconomic variables with subjective market
  expectations to fit the macroeconomic dynamics
  and the term structure.
• No reference to latent factors!

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Structure of the model

• This model uses a restricted time-varying
  structural VAR approach.
• In the (self-confirming) equilibrium the VAR
  corresponds to the reduced form of a standard
  rational expectations model.
• Allows for a structural interpretation of the
  parameters and a consistent modeling of the
  prices of risk (in common with Bekaert et al.,
  2005).
• On the disequilibrium path agents try to infer
  the long-run endpoints (inflation target,
  equilibrium real rate, ).
• Generates additional factors (very inert) not
  captured by a standard RE model
• Related to the learning literature (Orphanides
  and Williams, 2005 JEDC).
• Imposing no-arbitrage condition in as well as out
  of equilibrium generates consistent modeling of
  the term structure in terms of macroeconomic
  variables.

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A macroeconomic model of the term structure

• Structural equations Standard New-Keynesian
  framework.
• Learning Priors and Kalman filtering.
• Term structure model time-varying affine model.
  

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A structural macro-model of the term structure
New-Keynesian model

• A standard New-Keynesian RE model serves as
  benckmark model (Hördahl et al. (2005), Bekaert
  et al. (2005)).
• Uniqueness of the RE equilibrium is imposed on
  the model by imposing determinacy.
• The RE-equilibrium also serves as self-confirming
  equilibrium.
• Local stability is imposed with respect to the
  self-confirming equilibrium. SG-stability is
  imposed.
• The model serves as a locally attracting and
  unique benchmark for the actual dynamics under
  learning.

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A structural macro-model of the term structure
New-Keynesian model

• AS equation featuring endogenous inflation
  inertia
• IS equation featuring endogenous persistence
  through external habit formation
• Monetary policy rule
• Summary

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A structural macro-model of the term structure
Learning (Sargent and Williams, 2005)

• Beliefs of agents are modeled by
• (i) A perceived law of motion (based on a minimal
  state variable representation)
• (ii) Agents believe in macroeconomic instability.
  Attributing uncertainty to the long-run
  stochastic endpoint, we obtain a learning
  procedure
• The updating parameters ? are obtained from an
  approximate Kalman filtering procedure based on
  the agents priors.

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A structural macro-model of the term structure
Learning (Sargent and Williams, 2005)

• Beliefs of agents are modeled
• (i) In terms of a current perceived law of
  motion (within the Minimal State Variable
  models)
• Equivalently
• (ii) A set of priors concerning the (in)stability
  of the dynamics of the macroeconomy. The beliefs
  of agents are specified in a set of priors, V.

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A structural macro-model of the term structure
Learning (Sargent and Williams, 2005)

• Given the current perceived law of motion and a
  set of priors Vi , the MSE-optimal
  (equation-by-equation) filtering procedure is a
  Kalman filtering procedure (approximate Kalman
  filtering).
• As disussed in Sargent and Williams, 2005, the
  procedure specializes to constant gain RLS for a
  set of priors
• Yielding as solution

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A structural macro-model of the term structure
Learning

• Specializing to a local mean learning model
  agents believe in time-variation of the long-run
  endpoints.
• ?
• This model thus aims primarily at modeling the
  time-varying believes about the long run

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A structural macro-model of the term structure
Learning

• SG-stability conditions implies negativity
  conditions on the eigenvalues of the associated
  differential equation

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A structural macro-model of the term structure
Term structure

• No-arbitrage condition is imposed with respect to
  the risk adjusted subjective expectations
  operator (agents use their perceived law of
  motion).
• Assuming conditional normality of shocks, and the
  absence of arbitrage, the vector of yields, Y,
  can be written as an affine but time-varying
  function of the state variables.
• Assumption in pricing bonds, agents do not take
  into account the fact that their beliefs may
  change.

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A structural macro-model of the term structure
Term structure

• Term structure determination due to learning the
  yield curve is affine in the state vector but now
  with time-varying loadings.
• No-arbitrage ODEs
• with

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Model Summary
Perceived law of motion

Shocks
Agents beliefs
No-arbitrage loadings Ay(t),By(t)
Structural model (RE)
Yield curve
Economic state Xt
Updating of beliefs
Actual law of motion
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Estimation Methodology

• Using the actual law of motion, structural shocks
  can be identified and standard ML-techniques can
  be used.
• ALM dynamics can be identified under following
  assumptions
• Ex post, all (structural) parameter values and
  changes are known to the econometrician.
• The PLM dynamics are known to the econometrician.
  
• In line with the learning literature we lag the
  updating of the PLM by one period.

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Estimation Methodology

• Standard maximum likelihood is performed subject
  to a set of constraints
• The RE equilibrium exists and is uniquely
  determined. Determinacy implies that the Taylor
  principle needs to be satisfied.
• The self-confirming equilibrium is locally
  stable. This condition is imposed through the
  SG-stability conditions.
• Consistent modeling of the pricing kernel prices
  of risk are constrained by the structural
  parameters.
• All eigenvalues of the PLM dynamics are
  constrained to be smaller than 1 over the entire
  sample. This constraint ensures that subjective
  (long-run) expectations converge.
• Initial beliefs about the long run are estimated.
  
• Log-Likelihood

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Estimation methodology

• Actual law of motion (ALM) of macroeconomic
  variables
• Structural shocks

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Estimation methodology

• Actual law of motion (ALM) of yields
• Measurement error shocks

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Estimation methodology

• Combining the macroeconomic dynamics and the term
  structure
• Define
• Log-Likelihood

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Estimation Data

• Model is estimated on US data 1964Q1 - 1998Q4
• Quarter by quarter (CPI) inflation rates (in p.a.
  terms) are used. Source IMF Financial
  Statistics.
• CBO output gap measure is used (no real time
  data). Source Congressional Budget Office.
• FED rate is used as the policy rate. Source IMF
  Financial Statistics.
• Yields 2Q, 1, 2, 5 and 10 years. Source
  McCulloch and Kwon (1993), Bliss (1997) as
  reported by Duffee (2002).

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Data
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Data Summary statistics

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Overview of estimation results

• Estimation results
• Macroeconomic dynamics.
• Structural parameter estimates.
• Analysis of prediction errors.
• Estimates of historical policy stance.
• .
• Term structure dynamics.
• Analysis of term structure loadings.
• Decomposition of the term structure the impact
  of learning.
• Analysis of prediction errors.

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Estimation Results

• Estimation results are in line with existing
  studies as far as macroeconomic dynamics are
  concerned.
• Both forward and backward linkages are important.
  Especially in AS curve we find strong forward
  looking behavior.
• Interest rate effect on output is relatively
  large (relative to the literature).
• Feedback effect of output on inflation weak and
  imprecisely estimated.
• Taylor rule is recovered.
• Some model misspecification remains however in
  the form of autocorrelation in the residuals.
• Significant learning parameters.

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EstimationStructural model
Strongly forward-looking component
Large interest rate effect
Smoothing reasonable .75 Taylor-rule recovered
inflation 1.44, output. .44 Inflation targets
differ across chairmen
Large and significant learning parameters.
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Estimation Data, ALM and PLM dynamics
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Macroeconomic prediction errors
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Macroeconomic prediction errors
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Estimation Policy stance over time

• Estimated policy rule conforms to results in the
  literature.
• Taylor principle is satisfied with total
  sensitivity to inflation of about 1.44.
• Positive sensitivity to the CBO output gap of
  about 0.44.
• Significant interest rate smoothing relatively
  low 0.75. In line with the findings of
  Rudebusch, 2002, JME.
• Allowing for chairman-dependent inflation
  targets, differences in inflation targets are
  found (although insignificantly different)
• Martin 1964-1970 target 2.2
• Burns 1970-1978 target 7.9
• Miller 1978-1979 target 5,7
• Volcker,a 1979-1982 target 4.6
• Volcker,b 1982-1987 target 3.2
• Greenspan 1987-1999 target 3.5

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Estimation Policy stance over time

• Policy stance can be analyzed by computing the ex
  ante real interest rate based on (subjective)
  market expectations.
• Consistent with the recent learning literature,
  we find a weak policy stance during the Burns
  term.
• Volcker term shows a significant tightening of
  monetary policy.
• Policy stance during Greenspan era shows a
  moderate policy weak during the 1994 recession
  and tighter towards the end of the nineties.

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Estimation Policy stance relative to inflation
gap and output gap
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Estimation Term structure

• In general the term structure fit improved
  considerably relative to the RE macroeconomic
  model.
• Loadings of macroeconomic variables are related
  to slope and curvature factors
• Interest rate decreasing slope effect
• Inflation and output gap have hump-shaped
  loadings, with maximum impact on the 2-4 year
  maturity yields.
• No level factor found here!
• The level factor shifted to the state-independent
  loading (the A-loading).
• Although improving on the RE-model, the model is
  not fully satisfactory
• Relatively large measurement errors (relative to
  the latent factor models, reasonable relative to
  Bekaert et al. 2005 ).
• Some correlation is present in the prediction
  errors.

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Motivation Term structure fit RE
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Term structure decomposition
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Estimation time-invariant loadings
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Estimation Time-varying loading
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Prediction errors of term structure
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Prediction errors of term structure
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Cross-validation Survey expectations

• The improvement of term structure fit is due to
  the modeling of private agents expectations in a
  consistent learning framework. Do they make sense
  or are they simply an artifact to fit the term
  structure?
• Cross-validation of the estimated private
  expectations by comparing with survey
  expectations on inflation and interest rates.
• Survey of professional forecasters provides
  short-term survey expectations (1, 2, 3 and 4
  quarters ahead) starting 1981Q3 and long-run
  (average) forecasts since 1991.
• Combination of Blue chip, Livingston and SoPF
  average inflation over coming year and 10 years
  starting 1979Q4.
• Mean survey forecasts are used.

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Correlation Model-based versus survey
expectations of inflation.
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Inflation forecasts 1, 2, 3 and 4 quarters ahead
based on VAR
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Inflation forecasts 1, 2, 3 and 4 quarters ahead
based on model (learning II)
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Inflation forecasts 1 year and 10 year average
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Conclusions and further steps

• This paper presents a structural model for the
  macro-economy and the term structure taking
  explicitly into account learning (local mean
  models).
• Learning leads to significant changes relative to
  the benchmark RE model
• The representation of macroeconomic dynamics
  becomes time varying and is measured by the ALM.
• The affine term structure derived from
  no-arbitrage conditions becomes time varying and
  depends on market perceptions (PLM).

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Conclusions and further steps

• Estimation of the model leads to the following
  conclusions
• Estimates of the structural macroeconomic model
  are in line with previous findings (significant
  forward looking components, recovering the
  Taylor-rule).
• We find that the great inflation of the seventies
  is due to a weak policy stance in the seventies
  allowing inflationary shocks to feed in to
  long-run expectations.
• Learning dynamics are crucial in fitting the long
  end of the yield curve.
• Model is not fully satisfactory. Significant
  autocorrelation remains, possibly suggesting
  additional factors.
• The retrieved subjective expectations correlate
  strongly with survey expectations.

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Conclusions and further steps

• Further steps
• Sub-sample results indicate that the structural
  model has changed. We intend to investigate the
  implications for the sub-sample results.
• So far, learning was restricted to local mean
  learning. We intend to extend the learning
  procedure to VAR parameters as well.
• Updating data set.